A351616 -id:A351616 - cp's OEIS frontend

A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

0, 0, 0, 1, 1, 5, 4, 18, 19, 73, 73, 278, 283, 1076, 1090, 4125, 4183, 15939, 16105, 61628, 62170, 239388, 240907, 932230, 936447, 3641945, 3651618, 14262540, 14277519, 55987858, 55961118, 220223982, 219813564, 867835023, 865091976, 3425442681

Offset: 1

N. J. A. Sloane

nonn

This sequence gives the number of free polyominoes with symmetry group "R" in Redelmeier's notation. See his Tables 1 and 3, also the column "Rot" in Oliveira e Silva's table.

The rotation center of a polyomino with this symmetry may lie at the center of a square, the middle of an edge, or a vertex of a square. These subsets are enumerated by A351615, A234008 and A351616 respectively. - John Mason, Feb 17 2022, reformulated by Günter Rote, Oct 19 2023

			a(2) = 0 because the "domino" polyomino has symmetry group of order 4.
For n=3, the three-celled polyomino [ | | ] has group of order 4, and the polyomino
. [ ]
. [ | ]
has only reflective symmetry, so a(3) = 0.
a(4) = 1 because of (in Golomb's notation) the "skew tetromino".

S. W. Golomb, Polyominoes, Princeton Univ. Press, NJ, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 1..50
Tomás Oliveira e Silva, Enumeration of polyominoes
Tomás Oliveira e Silva, Numbers of polyominoes classified according to Redelmeier's symmetry classes (an extract from the previous link)
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.
Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351615, A234008, A351616.
Polyomino rings of length 2n with twofold rotational symmetry: A348402, A348403, A348404.

a(n) = A351615(n) + A234008(n/2) + A351616(n/2) for even n, otherwise a(n) = A351615(n). - John Mason, Feb 17 2022

Extended to n=28 by Tomás Oliveira e Silva
a(1)-a(3) prepended by Andrew Howroyd, Dec 04 2018
Edited by N. J. A. Sloane, Nov 28 2020
a(29)-a(36) from John Mason, Oct 16 2021

A351615 Number of free polyominoes with n squares, having 180-degree rotational symmetry about the center of a square, and no other symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 4, 0, 19, 1, 73, 4, 283, 25, 1090, 106, 4183, 463, 16105, 1892, 62170, 7752, 240907, 31212, 936447, 125609, 3651618, 503165, 14277519, 2014826, 55961118, 8058790, 219813564, 32231897, 865091976, 128897247, 3410498446, 515554767, 13466262587, 2062445063, 53245595410, 8252868076, 210800433323

Offset: 1

John Mason, Feb 15 2022

nonn

The sequence enumerates a subset of the polyominoes enumerated by A006747.

			a(5)=1 because of:
   OO
   O
  OO

John Mason, Table of n, a(n) for n = 1..50

Cf. A006747, A234008, A351616.

More terms from John Mason, Apr 15 2023

A331621 Number of distinct structures that can be made from n cubes without overhangs.

Original entry on oeis.org

1, 1, 2, 4, 12, 35, 129, 495, 2101, 9154, 41356, 189466, 880156, 4120515, 19425037, 92038062, 438030079, 2092403558, 10027947217, 48198234188, 232261124908, 1121853426115, 5430222591596

Offset: 0

Nicholas A. Kennedy, Jan 22 2020

This is an extension of the free polyominoes (A000105) to the third dimension. Structures are considered equivalent if they can be mapped into each other by reflection in a vertical plane or rotation around the vertical axis. They are not equivalent if they can only be mapped onto each other by rotation around an axis parallel to the horizontal plane.
From John Mason, Mar 03 2025: (Start)
Equivalently, the sequence enumerates inscribed polyominoes that have a positive integer in each square, such that the size of the polyomino is considered to be the sum of the integers.
Examples.
  Size 1:
  +-+
  |1|
  +-+
.
  Size 2:
  +-+-+ +-+
  |1|1| |2|
  +-+-+ +-+
.
  Size 3:
  +-+-+-+ +-+-+ +-+-+ +-+
  |1|1|1| |2|1| |1|1| |3|
  +-+-+-+ +-+-+ +-+-+ +-+
                  |1|
                  +-+
(End)

			For n = 0, one (the empty) structure is possible.
For n = 1, only one structure is possible, a single cube.
For n = 2, two structures are possible: two cubes one on top of the other, and two next to each other.
For n = 3, four structures are possible: an L shape with the L oriented vertically, an L shape with the L laid flat, a structure with 3 cubes stacked on top of each other and a structure with 3 cubes laid flat.
For n = 4, there are 12 possible distinct structures made from 4 cubes without overhangs. These include 1 structure that is 4 cubes tall, 1 structure that is 3 cubes tall, 5 structures that are 2 cubes tall and 5 that are 1 cube tall.

Arnauld Chevallier, Javascript Node.JS program
Nicholas A. Kennedy, Buildings made from cubes, Code Golf Stack Exchange 2020
Nicholas A. Kennedy, Images of possible structures made from 4 cubes
Miles, Java program

Extension to the third dimension of A000105.

Cf. A006749, A007318, A259090, A349329, A351616, A234008.

Java
```
See Miles link
```
JavaScript
```
See Arnauld Chevallier link
```

From John Mason, Mar 03 2025: (Start)
Define a(n)=f(n)+g(n) where f(n) enumerates the structures having an asymmetrical base, and g(n) enumerates the structures having a symmetrical base.
Then for n>=4, f(n) = Sum_{i=4..n} ((A006749(i)*C(n-1,i-1)), and g(n) < Sum_{i=1..n} ((A259090(i)*C(n-1,i-1)).
For structures having a base with: reflective orthogonal symmetry about an axis that passes through cell vertices, 180 degree rotational symmetry about a point at a cell vertex or midway along an edge, the number of structures is (1) for odd n: Sum_{i=1..n} ((S(i)*C(n-1,i-1)) / 2, and (2) for even n: Sum_{i=1..n} ((S(i)*(C(n-1,i-1)+C(n/2-1,i/2-1)) / 2, for S(n) = A349329(n), A351616(n) and A234008(n) respectively.
As almost all polyominoes are asymmetrical, a(n)/f(n) tends to 1. (End)

a(15)-a(22) from John Mason, Mar 03 2025

cp's OEIS Frontend

A006747 Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A351615 Number of free polyominoes with n squares, having 180-degree rotational symmetry about the center of a square, and no other symmetry.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A331621 Number of distinct structures that can be made from n cubes without overhangs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

JavaScript

Formula

Extensions